Advanced Search
Article Contents
Article Contents

On well-posedness of vector-valued fractional differential-difference equations

  • * Corresponding author: Carlos Lizama

    * Corresponding author: Carlos Lizama 

C. Lizama has been partially supported by DICYT, Universidad de Santiago de Chile and FONDECYT 1180041. L. Abadias and P. J. Miana have been partially supported by Project MTM2016-77710-P, DGI-FEDER, of the MCYTS, and Project E-64 D.G. Aragón. M. P. Velasco has been partially supported by Project ESP2016-79135-R of the MCYTS

Abstract Full Text(HTML) Related Papers Cited by
  • We develop an operator-theoretical method for the analysis on well posedness of partial differential-difference equations that can be modeled in the form

    $ \begin{equation*} (*) \left\{\begin{array}{rll} \Delta^{\alpha} u(n) & = & Au(n+2) + f(n,u(n)), \quad n \in \mathbb{N}_0, \,\, 1< \alpha \leq 2; \\ u(0) & = & u_0;\\ u(1) & = & u_1, \end{array}\right. \end{equation*} $

    where $ A $ is a closed linear operator defined on a Banach space $ X $. Our ideas are inspired on the Poisson distribution as a tool to sampling fractional differential operators into fractional differences. Using our abstract approach, we are able to show existence and uniqueness of solutions for the problem (*) on a distinguished class of weighted Lebesgue spaces of sequences, under mild conditions on sequences of strongly continuous families of bounded operators generated by $ A, $ and natural restrictions on the nonlinearity $ f $. Finally we present some original examples to illustrate our results.

    Mathematics Subject Classification: Primary: 35R11; Secondary: 39A14, 47D06.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] L. AbadiasM. De León and J. L. Torrea, Non-local fractional derivatives. Discrete and continuous, J. Math. Anal. Appl., 449 (2017), 734-755.  doi: 10.1016/j.jmaa.2016.12.006.
    [2] L. Abadias and C. Lizama, Almost automorphic mild solutions to fractional partial difference-differential equations, Appl. Anal., 95 (2016), 1347-1369.  doi: 10.1080/00036811.2015.1064521.
    [3] L. AbadiasC. LizamaP. J. Miana and M. P. Velasco, Cesàro sums and algebra homomorphisms of bounded operators, Israel J. Math., 216 (2016), 471-505.  doi: 10.1007/s11856-016-1417-3.
    [4] L. Abadias and P. J. Miana, A subordination principle on Wright functions and regularized resolvent families, J. Funct. Spaces, Art., 2015 (2015), ID 158145, 9 pp. doi: 10.1155/2015/158145.
    [5] T. Abdeljawad and F. M. Atici, On the definitions of nabla fractional operators, Abstr. Appl. Anal., 2012 (2012), Art. ID 406757, 13 pp. doi: 10.1155/2012/406757.
    [6] G. AkrivisB. Li and C. Lubich, Combining maximal regularity and energy estimates for the discretizations of quasilinear parabolic equations, Math. Comp., 86 (2017), 1527-1552.  doi: 10.1090/mcom/3228.
    [7] W. Arendt, C. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics. vol. 96. Birkhäuser, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.
    [8] F. M. Atici and P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Difference Equ., 2 (2007), 165-176. 
    [9] F. M. Atici and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electr. J. Qual. Th. Diff. Equ., 3 (2009), 1-12.  doi: 10.14232/ejqtde.2009.4.3.
    [10] F. M. Atici and S. Sengül, Modeling with fractional difference equations, J. Math. Anal. Appl., 369 (2010), 1-9.  doi: 10.1016/j.jmaa.2010.02.009.
    [11] F. M. Atici and P. W. Eloe, Linear systems of fractional nabla difference equations, Rocky Mountain J. Math., 41 (2011), 353-370.  doi: 10.1216/RMJ-2011-41-2-353.
    [12] Y. BaiD. Baleanu and G. C. Wu, Existence and discrete approximation for optimization problems governed by fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 338-348.  doi: 10.1016/j.cnsns.2017.11.009.
    [13] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, 2012. doi: 10.1142/9789814355216.
    [14] E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001.
    [15] J. CermákT. Kisela and L. Nechvátal, Stability and asymptotic properties of a linear fractional difference equation, Adv. Difference Equ., 122 (2012), 1-14.  doi: 10.1186/1687-1847-2012-122.
    [16] J. CermákT. Kisela and L. Nechvátal, Stability regions for linear fractional differential systems and their discretizations, Appl. Math. Comput., 219 (2013), 7012-7022.  doi: 10.1016/j.amc.2012.12.019.
    [17] E. CuestaC. Lubich and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations, Math. Comp., 75 (2006), 673-696.  doi: 10.1090/S0025-5718-06-01788-1.
    [18] E. Cuesta and C. Palencia, A numerical method for an integro-differential equation with memory in Banach spaces: Qualitative properties, SIAM J. Numer. Anal., 41 (2003), 1232-1241.  doi: 10.1137/S0036142902402481.
    [19] E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin. Dyn. Syst., 2007, Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, suppl., 277–285.
    [20] I. K. DassiosD. I. Baleanu and G. I. Kalogeropoulos, On non-homogeneous singular systems of fractional nabla difference equations, Appl. Math. Comput., 227 (2014), 112-131.  doi: 10.1016/j.amc.2013.10.090.
    [21] K. J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000.
    [22] T. Kemmochi and N. Saito, Discrete maximal regularity and the finite element method for parabolic equations, Num. Math., 138 (2018), 905-937.  doi: 10.1007/s00211-017-0929-z.
    [23] V. KeyantuoC. Lizama and M. Warma, Spectral criteria for solvability of boundary value problems and positivity of solutions of time-fractional differential equations, Abstr. Appl. Anal., 2013 (2013), 1-11.  doi: 10.1155/2013/614328.
    [24] C. Lizama, The Poisson distribution, abstract fractional difference equations, and stability, Proc. Amer. Math. Soc., 145 (2017), 3809-3827.  doi: 10.1090/proc/12895.
    [25] C. Lizama, $l_p$-maximal regularity for fractional difference equations on UMD spaces, Math. Nach., 288 (2015), 2079-2092.  doi: 10.1002/mana.201400326.
    [26] C. Lizama and M. P. Velasco, Weighted bounded solutions for a class of nonlinear fractional equations, Fract. Calc. Appl. Anal., 19 (2016), 1010-1030.  doi: 10.1515/fca-2016-0055.
    [27] C. Lizama and M. Murillo-Arcila, $\ell_p$-maximal regularity for a class of fractional difference equations on $UMD$ spaces: The case $1 < \alpha \leq 2,$, Banach J. Math. Anal., 11 (2017), 188-206.  doi: 10.1215/17358787-3784616.
    [28] C. Lizama and M. Murillo-Arcila, Maximal regularity in $\ell_p$ spaces for discrete time fractional shifted equations, J. Differential Equations, 263 (2017), 3175-3196.  doi: 10.1016/j.jde.2017.04.035.
    [29] K. S. Miller and B. Ross, Fractional difference calculus, In: Univalent Functions, Fractional Calculus, and Their Applications (Kóriyama, 1988), Horwood, Chichester, (1989), 139–152.
    [30] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series. Elementary functions, Vol. 1. Gordon and Breach Science Publishers, New York, 1986.
    [31] A. M. SinclairContinuous Semigroups in Banach Algebras, London Mathematical Society, Lecture Notes Series 63, Cambridge University Press, New York, 1982. 
    [32] C. C. Travis and G. F. Webb, Compactness, regularity, and uniform continuity properties of strongly continuous cosine families, Houston Journal of Mathematics, 3 (1977), 555-567. 
    [33] L. W. Weis, A generalization of the Vidav-Jorgens perturbation theorem for semigroups and its application to Transport Theory, J. Math. Anal. Appl., 129 (1988), 6-23.  doi: 10.1016/0022-247X(88)90230-2.
    [34] G. C. WuD. Baleanu and L. L. Huang, Novel Mittag-Leffler stability of linear fractional delay difference equations with impulse, Appl. Math. Lett., 82 (2018), 71-78.  doi: 10.1016/j.aml.2018.02.004.
    [35] G. C. Wu, D. Baleanu and H. P. Xie, Riesz Riemann–Liouville difference on discrete domains, Chaos, 26 (2016), 084308, 5 pp. doi: 10.1063/1.4958920.
    [36] G. C. Wu and D. Baleanu, Discrete chaos in fractional delayed logistic maps, Nonlinear Dynam., 80 (2016), 1697-1703.  doi: 10.1007/s11071-014-1250-3.
    [37] G. C. Wu and D. Baleanu, Stability analysis of impulsive fractional difference equations, Fract. Calc. Appl. Anal., 21 (2018), 354-375.  doi: 10.1515/fca-2018-0021.
    [38] A. ZygmundTrigonometric Series, 2nd ed. Vols. Ⅰ, Ⅱ, Cambridge University Press, New York, 1959. 
  • 加载中

Article Metrics

HTML views(865) PDF downloads(226) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint